Distance between a line and a point calculator

Find the distance from line $ L: y = x $ to the point $ \left(0,~0\right) $.

solution

The distance between the line and the point is:

$$ d = 0 $$

explanation

The distance from the point $ (x_0, y_0) $ to the line $ Ax + By + C = 0 $ is given by:

$$ d = \frac{ \left| Ax_0 + B_yo + C \right| }{ \sqrt{A^2 + B^2}} $$

To apply this formula, we first need to express the line in standard form

$$\begin{aligned} y & = x \\x-y&=0\end{aligned}$$

After substituting: $ A = 1 $ , $ B = -1 $ , $ C = 0 $ , $ x_0 = 0 $ and $ y_0 = 0 $ we have:

$$ \begin{aligned} d =& \frac{ \left| 1\cdot0 +\left(-1\right)\cdot0 + 0 \right| }{\sqrt{ 1^2 + (-1)^2}} = \\ d =& \frac{ \left| 0 + 0 + 0 \right| }{\sqrt{ 1 + 1}} = \\ d =& \frac{ \left| 0 \right| }{\sqrt{ 2}} = \\ d =& \frac{ 0 }{ \sqrt{ 2 } } = \\ d =& 0 \end{aligned} $$